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Properties

Label 2023.1.c.e
Level 20232023
Weight 11
Character orbit 2023.c
Analytic conductor 1.0101.010
Analytic rank 00
Dimension 44
Projective image D5D_{5}
CM discriminant -119
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2023,1,Mod(1735,2023)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2023, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2023.1735");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: N N == 2023=7⋅172 2023 = 7 \cdot 17^{2}
Weight: k k == 1 1
Character orbit: [χ][\chi] == 2023.c (of order 22, degree 11, minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 1.009608520561.00960852056
Analytic rank: 00
Dimension: 44
Coefficient field: Q(i,5)\Q(i, \sqrt{5})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4+3x2+1 x^{4} + 3x^{2} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 119)
Projective image: D5D_{5}
Projective field: Galois closure of 5.1.14161.1
Artin image: C4×D5C_4\times D_5
Artin field: Galois closure of Q[x]/(x20−⋯ )\mathbb{Q}[x]/(x^{20} - \cdots)

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the qq-expansion are expressed in terms of a basis 1,β1,β2,β31,\beta_1,\beta_2,\beta_3 for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q−β2q2−β1q3−β2q4+(β3+β1)q5+(β3−β1)q6+β3q7+q8+β2q9−β3q10+(β3−β1)q12+⋯+β2q98+O(q100) q - \beta_{2} q^{2} - \beta_1 q^{3} - \beta_{2} q^{4} + (\beta_{3} + \beta_1) q^{5} + (\beta_{3} - \beta_1) q^{6} + \beta_{3} q^{7} + q^{8} + \beta_{2} q^{9} - \beta_{3} q^{10} + (\beta_{3} - \beta_1) q^{12}+ \cdots + \beta_{2} q^{98}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 4q+2q2+2q4+4q8−2q9+4q15−6q18−2q21−2q25+2q30−4q32−2q35−6q36−6q42+2q43−4q49+4q50+2q53+2q60−2q64+⋯−2q98+O(q100) 4 q + 2 q^{2} + 2 q^{4} + 4 q^{8} - 2 q^{9} + 4 q^{15} - 6 q^{18} - 2 q^{21} - 2 q^{25} + 2 q^{30} - 4 q^{32} - 2 q^{35} - 6 q^{36} - 6 q^{42} + 2 q^{43} - 4 q^{49} + 4 q^{50} + 2 q^{53} + 2 q^{60} - 2 q^{64}+ \cdots - 2 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x4+3x2+1 x^{4} + 3x^{2} + 1 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== ν2+1 \nu^{2} + 1 Copy content Toggle raw display
β3\beta_{3}== ν3+2ν \nu^{3} + 2\nu Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β2−1 \beta_{2} - 1 Copy content Toggle raw display
ν3\nu^{3}== β3−2β1 \beta_{3} - 2\beta_1 Copy content Toggle raw display

Display βi\beta_i in terms of νj\nu^j

Character values

We give the values of χ\chi on generators for (Z/2023Z)×\left(\mathbb{Z}/2023\mathbb{Z}\right)^\times.

nn 290290 17371737
χ(n)\chi(n) −1-1 11

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\iota_m(a_n) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   ιm(ν)\iota_m(\nu) a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
1735.1
0.618034i
− 0.618034i
1.61803i
− 1.61803i
−0.618034 − 0.618034i −0.618034 1.61803i 0.381966i 1.00000i 1.00000 0.618034 − 1.00000i
1735.2 −0.618034 0.618034i −0.618034 − 1.61803i − 0.381966i − 1.00000i 1.00000 0.618034 1.00000i
1735.3 1.61803 − 1.61803i 1.61803 0.618034i − 2.61803i − 1.00000i 1.00000 −1.61803 1.00000i
1735.4 1.61803 1.61803i 1.61803 − 0.618034i 2.61803i 1.00000i 1.00000 −1.61803 − 1.00000i
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
119.d odd 2 1 CM by Q(−119)\Q(\sqrt{-119})
7.b odd 2 1 inner
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2023.1.c.e 4
7.b odd 2 1 inner 2023.1.c.e 4
17.b even 2 1 inner 2023.1.c.e 4
17.c even 4 1 119.1.d.a ✓ 2
17.c even 4 1 119.1.d.b yes 2
17.d even 8 4 2023.1.f.b 8
17.e odd 16 8 2023.1.l.b 16
51.f odd 4 1 1071.1.h.a 2
51.f odd 4 1 1071.1.h.b 2
68.f odd 4 1 1904.1.n.a 2
68.f odd 4 1 1904.1.n.b 2
85.f odd 4 1 2975.1.b.a 4
85.f odd 4 1 2975.1.b.b 4
85.i odd 4 1 2975.1.b.a 4
85.i odd 4 1 2975.1.b.b 4
85.j even 4 1 2975.1.h.c 2
85.j even 4 1 2975.1.h.d 2
119.d odd 2 1 CM 2023.1.c.e 4
119.f odd 4 1 119.1.d.a ✓ 2
119.f odd 4 1 119.1.d.b yes 2
119.l odd 8 4 2023.1.f.b 8
119.m odd 12 2 833.1.h.a 4
119.m odd 12 2 833.1.h.b 4
119.n even 12 2 833.1.h.a 4
119.n even 12 2 833.1.h.b 4
119.p even 16 8 2023.1.l.b 16
357.l even 4 1 1071.1.h.a 2
357.l even 4 1 1071.1.h.b 2
476.k even 4 1 1904.1.n.a 2
476.k even 4 1 1904.1.n.b 2
595.l even 4 1 2975.1.b.a 4
595.l even 4 1 2975.1.b.b 4
595.r even 4 1 2975.1.b.a 4
595.r even 4 1 2975.1.b.b 4
595.u odd 4 1 2975.1.h.c 2
595.u odd 4 1 2975.1.h.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
119.1.d.a ✓ 2 17.c even 4 1
119.1.d.a ✓ 2 119.f odd 4 1
119.1.d.b yes 2 17.c even 4 1
119.1.d.b yes 2 119.f odd 4 1
833.1.h.a 4 119.m odd 12 2
833.1.h.a 4 119.n even 12 2
833.1.h.b 4 119.m odd 12 2
833.1.h.b 4 119.n even 12 2
1071.1.h.a 2 51.f odd 4 1
1071.1.h.a 2 357.l even 4 1
1071.1.h.b 2 51.f odd 4 1
1071.1.h.b 2 357.l even 4 1
1904.1.n.a 2 68.f odd 4 1
1904.1.n.a 2 476.k even 4 1
1904.1.n.b 2 68.f odd 4 1
1904.1.n.b 2 476.k even 4 1
2023.1.c.e 4 1.a even 1 1 trivial
2023.1.c.e 4 7.b odd 2 1 inner
2023.1.c.e 4 17.b even 2 1 inner
2023.1.c.e 4 119.d odd 2 1 CM
2023.1.f.b 8 17.d even 8 4
2023.1.f.b 8 119.l odd 8 4
2023.1.l.b 16 17.e odd 16 8
2023.1.l.b 16 119.p even 16 8
2975.1.b.a 4 85.f odd 4 1
2975.1.b.a 4 85.i odd 4 1
2975.1.b.a 4 595.l even 4 1
2975.1.b.a 4 595.r even 4 1
2975.1.b.b 4 85.f odd 4 1
2975.1.b.b 4 85.i odd 4 1
2975.1.b.b 4 595.l even 4 1
2975.1.b.b 4 595.r even 4 1
2975.1.h.c 2 85.j even 4 1
2975.1.h.c 2 595.u odd 4 1
2975.1.h.d 2 85.j even 4 1
2975.1.h.d 2 595.u odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S1new(2023,[χ])S_{1}^{\mathrm{new}}(2023, [\chi]):

T22−T2−1 T_{2}^{2} - T_{2} - 1 Copy content Toggle raw display
T11 T_{11} Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T2−T−1)2 (T^{2} - T - 1)^{2} Copy content Toggle raw display
33 T4+3T2+1 T^{4} + 3T^{2} + 1 Copy content Toggle raw display
55 T4+3T2+1 T^{4} + 3T^{2} + 1 Copy content Toggle raw display
77 (T2+1)2 (T^{2} + 1)^{2} Copy content Toggle raw display
1111 T4 T^{4} Copy content Toggle raw display
1313 T4 T^{4} Copy content Toggle raw display
1717 T4 T^{4} Copy content Toggle raw display
1919 T4 T^{4} Copy content Toggle raw display
2323 T4 T^{4} Copy content Toggle raw display
2929 T4 T^{4} Copy content Toggle raw display
3131 T4+3T2+1 T^{4} + 3T^{2} + 1 Copy content Toggle raw display
3737 T4 T^{4} Copy content Toggle raw display
4141 T4+3T2+1 T^{4} + 3T^{2} + 1 Copy content Toggle raw display
4343 (T2−T−1)2 (T^{2} - T - 1)^{2} Copy content Toggle raw display
4747 T4 T^{4} Copy content Toggle raw display
5353 (T2−T−1)2 (T^{2} - T - 1)^{2} Copy content Toggle raw display
5959 T4 T^{4} Copy content Toggle raw display
6161 T4+3T2+1 T^{4} + 3T^{2} + 1 Copy content Toggle raw display
6767 (T2+T−1)2 (T^{2} + T - 1)^{2} Copy content Toggle raw display
7171 T4 T^{4} Copy content Toggle raw display
7373 T4+3T2+1 T^{4} + 3T^{2} + 1 Copy content Toggle raw display
7979 T4 T^{4} Copy content Toggle raw display
8383 T4 T^{4} Copy content Toggle raw display
8989 T4 T^{4} Copy content Toggle raw display
9797 T4+3T2+1 T^{4} + 3T^{2} + 1 Copy content Toggle raw display
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